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CU-Boulder PHYS 7450 - Beware of density dependent pair potentials

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTERJ. Phys.: Condens. Matter 14 (2002) 9187–9206 PII: S0953-8984(02)36566-4Beware of density dependent pair potentialsAALouisDepartment of Chemistry, Lensfield Rd, Cambridge CB2 1EW, UKReceived 2 May 2002, in final form 8 July 2002Published 27 September 2002Onlineatstacks.iop.org/JPhysCM/14/9187AbstractDensity (or state) dependent pair potentials arise naturally from coarse-grainingprocedures in many areas of condensed matter science. However, correctlyusing them to calculate physical properties of interest is subtle and cannot beuncoupled from the route by which they were derived. Furthermore, there isusually no unique way to coarse-grain to an effective pair potential. Even forsimple systems like liquid argon, the pair potential that correctly reproduces thepair structure will not generate the right virial pressure. Ignoring these issues innaive applications of density dependent pair potentials can lead to an apparentdependence of thermodynamic propertiesonthe ensemble within which theyare calculated, as well as other inconsistencies. These concepts are illustrated byseveral pedagogical examples, including effective pair potentials for systemswith many-body interactions, and the mapping of charged (Debye–H¨uckel)and uncharged (Asakura–Oosawa) two-component systems onto effective one-component ones. The differences between the problems of transferability andrepresentability for effective potentials are also discussed.1. IntroductionNo known materialsexistinnature whose behaviour can be completely captured by pairpotentials alone. Even the properties of a noble gas like argon have a finite contributionfrom three-body Axilrod–Teller triple-dipole interactions [1]. Thus, the pair potentials usedto describe condensed matter systems always arise from coarse-graining procedures, whereasubset of the degrees of freedom of the full (quantum) statistical mechanical system areintegrated out. In the aforementioned example, integrating the three-body interactions overangular coordinates results in effective parameters for the Lennard-Jones (LJ) pair potential,which will depend on state. Similarly, in metallic systems, integrating out the free electronsleads to a configuration independent volume term and pair potentials that depend on the globaldensity [2]. Alternative coarse-graining procedures for metals such as the embedded atommethod [3, 4], effective medium theory [5], Finnis–Sinclair potentials [6] or glue potentials [7]result in a local density or environment dependence.Coarse-graining methods are also crucial to deriving tractable statistical mechanicaltreatments of soft-matter systems, where a large number of different length and timescales0953-8984/02/409187+20$30.00 © 2002 IOP Publishing Ltd Printed in the UK 91879188 AALouismay coexist. An increasingly popular coarse-graining technique consists of deriving effectivepotentials and exploiting their analogy with well studied simple atomic or molecular systems toextract phase behaviour and correlations [8–10]. Again, these effective interactions are oftenreduced to an approximate pairwise description with parameters that depend on state.Direct inversion from experimental structure factors are another way to derive theparameters for effective pairwise potentials [11, 12]. These almost always show a dependenceon state, especially for the case of soft-matter systems. This is not surprising, of course, sinceone can easily imagine thattheinteractions between two effective particles depend on theoverall density. For example, changing the concentration of a micellar solution may affect theinternal structure of the micelles, which in turnleadstoadensitydependence of the effectivepair interaction between the particles.That an effective pair potential derived in one context does not always perform well inanother is well known, and usually categorized as a problem of transferability.Forexample,if the parameters of an effective pair potential depend on density, then a parametrization of thepotential at ρ1is not the same as the one needed at a different density ρ2—the potential at ρ1is not transferable to the state point at ρ2.Because it is usually hard to derive an explicit statedependence, a given potential is often used for state points close enough to the one for whichit was parametrized that transferability problems are not deemed to be important. In this casethe potential is usually treated as if it were independent of state.What I will endeavour to show in the present paper is that there are deeper problemsassociated with the use of effective pair potentials, even when the problem of transferabilityappears to be solved. These include problems of representability:atagivenstate point, nosingle pair potential may exist that can capture all the properties of a given material. Theparticular example of state dependence studied is pair potentials that depend on the globaldensity ρ as v(r;ρ)1.The paper’s focus is partially pedagogical. For that reason rather simplemodels are treated, with a special emphasis on the liquid phase. Some of these results havealready appeared in one form or another in the literature, and will be briefly reviewed.The paper is organized as follows: section 2 describes the apparent inconsistencies thatarise between the virial and compressibility routes to thermodynamics for a simple densitydependent pair potential v(r;ρ).Section 3 discusses the effective pair potentials that resultfrom integrating out third and higher order many-body interactions. The effective pair potentialthat correctly describes the excess internal energy is shown to be different from the one thatcorrectly describes the pair structure. These points are illustrated with a specific applicationfrom polymer solutions. In section 4 the McMillan–Mayer [13] tracing-out procedure isanalysed for an exactly solvable lattice version of the Asakura–Oosawa (AO) [14] model.While this procedure maps onto a useful effective one-component picture in the semi-grandensemble, integrating out the smaller particles in a canonical ensemble does not lead to aneffective Hamiltonian decomposable as a sum over independent interactions. For chargedsystems the canonical ensemble is the natural choice to integrate out microscopic co- andcounterions. Again, apparent ambiguities arise when the density dependent Debye–H¨uckelpotential is used to derive thermodynamics. Finally, conclusions


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